L(s) = 1 | + 3·5-s − 2·7-s + 6·11-s + 5·13-s − 3·17-s − 2·19-s − 6·23-s + 4·25-s + 3·29-s + 4·31-s − 6·35-s + 5·37-s − 6·41-s + 10·43-s − 3·49-s − 6·53-s + 18·55-s + 12·59-s + 5·61-s + 15·65-s − 2·67-s − 6·71-s − 73-s − 12·77-s + 10·79-s − 9·85-s − 3·89-s + ⋯ |
L(s) = 1 | + 1.34·5-s − 0.755·7-s + 1.80·11-s + 1.38·13-s − 0.727·17-s − 0.458·19-s − 1.25·23-s + 4/5·25-s + 0.557·29-s + 0.718·31-s − 1.01·35-s + 0.821·37-s − 0.937·41-s + 1.52·43-s − 3/7·49-s − 0.824·53-s + 2.42·55-s + 1.56·59-s + 0.640·61-s + 1.86·65-s − 0.244·67-s − 0.712·71-s − 0.117·73-s − 1.36·77-s + 1.12·79-s − 0.976·85-s − 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.223231888\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.223231888\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.604404195328056888635195624166, −9.007248355470093411359381448674, −8.298117991686099747490643532078, −6.76359618838495100744614032144, −6.29860250743506028393247154678, −5.87100059007609562623927091882, −4.38712587564334162926568406315, −3.59193545168410033442089288096, −2.26841618983682361893117043506, −1.22349289581102954381038244904,
1.22349289581102954381038244904, 2.26841618983682361893117043506, 3.59193545168410033442089288096, 4.38712587564334162926568406315, 5.87100059007609562623927091882, 6.29860250743506028393247154678, 6.76359618838495100744614032144, 8.298117991686099747490643532078, 9.007248355470093411359381448674, 9.604404195328056888635195624166